This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A228576 A triangle formed like generalized Pascal's triangle. The rule is T(n,k) = 2*T(n-1,k-1) + T(n-1,k), the left border is n and the right border is n^2 instead of 1. 19
 0, 1, 1, 2, 3, 4, 3, 7, 10, 9, 4, 13, 24, 29, 16, 5, 21, 50, 77, 74, 25, 6, 31, 92, 177, 228, 173, 36, 7, 43, 154, 361, 582, 629, 382, 49, 8, 57, 240, 669, 1304, 1793, 1640, 813, 64, 9, 73, 354, 1149, 2642, 4401, 5226, 4093, 1690, 81, 10, 91, 500, 1857, 4940 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Boris Putievskiy, Rows n = 1..140 of triangle, flattened Rely Pellicer, David Alvo, Modified Pascal Triangle and Pascal Surfaces p.4 Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO] Rattanapol Wasutharat, Kantaphon Kuhapatanakul, The Generalized Pascal-Like Triangle and Applications Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 41, pp. 1989 - 1992 FORMULA T(n,0)=n^2, T(0,k) = k, T(n, k) = 2*T(n-1, k-1) + T(n-1, k) for n,k >=0. Closed-form formula for generalized Pascal's triangle. Let a,b be any numbers. The rule is T(n, k) = a*T(n-1, k-1) + b*T(n-1, k) for n,k >0. Let L(m) and R(m) be the left border and the right border generalized Pascal's triangle, respectively. As table read by antidiagonals T(n,k)=sum_{m1=1..n}a^(n-m1)*b^k*R(m1)*C(n+k-m1-1,n-m1)+sum_{m2=1..k}a^n*b^(k-m2)*L(m2)*C(n+k-m2-1,k-m2); n,k >=0. As linear sequence a(n)=sum_{m1=1..i}a^(i-m1)*b^j*R(m1)*C(i+j-m1-1,i-m1)+sum_{m2=1..j}a^i*b^(j-m2)*L(m2)*C(i+j-m2-1,j-m2), where  i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2); n>0. Some special cases. If a=b=1,  then the closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. If a=0, then as table read by antidiagonals  T(n,k)=b*R(n), as linear sequence a(n)=b*R(i), where  i=n-t*(t+1)/2-1, t=floor((-1+sqrt(8*n-7))/2); n>0. The sequence a(n) is the reluctant sequence of sequence b*R(n) - a(n) is triangle array read by rows: row number k coincides with first k elements of the sequence b*R(n). Similarly for b=0, we get T(n,k)=a*L(k). For this sequence  L(m)=m and R(m)=m^2, a=2, b=1. As table  read by antidiagonals T(n,k)=sum_{m1=1..n}2^(n-m1)*m1**2*C(n+k-m1-1,n-m1)+sum_{m2=1..k}2^n*m2*C(n+k-m2-1,k-m2); n,k >=0. As linear sequence a(n)=sum_{m1=1..i}2^(i-m1)*m1**2*C(i+j-m1-1,i-m1)+sum_{m2=1..j}2^i*m2*C(i+j-m2-1,j-m2), where  i=n-t*(t+1)/2-1, j=(t*t+3*t+4)/2-n-1, t=floor((-1+sqrt(8*n-7))/2); n>0. EXAMPLE The start of the sequence as triangle array read by rows: 0; 1,1; 2,3,4; 3,7,10,9; 4,13,24,29,16; 5,21,50,77,74,25; ... MAPLE T := proc(n, k) option remember; if k = 0 then RETURN(n) fi; if k = n then RETURN(n^2) fi; 2*T(n-1, k-1) + T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..9);  # Peter Luschny, Aug 26 2013 MATHEMATICA T[n_, 0] := n; T[n_, n_] := n^2; T[n_, k_] := T[n, k] = 2*T[n-1, k-1]+T[n-1, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014 *) CROSSREFS Cf. We denote generalized Pascal's like triangle with coefficients a, b and with L(n) on the left border and R(n) on the right border by (a,b,L(n),R(n)). The list of sequences for (1,1,L(n),R(n)) see A228196; A038207 (1,2,2^n,1), A105728 (1, 2, 1, n+1), A112468 (1,-1,1,1),  A112626 (1,2,3^n,1), A119258 (2,1,1,1), A119673 (3,1,1,1), A119725 (3,2,1,1),  A119726 (4,2,1,1), A119727 (5,2,1,1), A209705 (2,1,n+1,0); A002061 (column 2), A000244 (sums of rows r of triangle array - (r-2)(r+1)/2). Sequence in context: A048259 A324150 A198461 * A324195 A211507 A295368 Adjacent sequences:  A228573 A228574 A228575 * A228577 A228578 A228579 KEYWORD nonn,tabl AUTHOR Boris Putievskiy, Aug 26 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 22 00:49 EDT 2019. Contains 328315 sequences. (Running on oeis4.)